Problem 12 - Precalculus Essentials

Question

The functions are all one-to-one. For each function, a. Find an equation for \(f^{-1}(x)\), the inverse function. b. Verify that your equation is correct by showing that $$f\left(f^{-1}(x)\right)=x \text { and } f^{-1}(f(x))=x$$ $$f(x)=x+5$$

Step-by-Step Solution

Answer

Hence, the inverse of the function \(f(x) = x + 5\) is \(f^{-1}(x) = x - 5\), and it has been verified to be correct by satisfying the conditions \(f(f^{-1}(x)) = x\) and \(f^{-1}(f(x)) = x\).

Step 1: Finding the Inverse Function

To find the inverse of a function, one must change \(f(x)\) to \(y\) and swap \(x\) and \(y\). For \(y = x + 5\), swapping \(x\) and \(y\) gives \(x = y + 5\). Solving for \(y\), the inverse function \(f^{-1}(x)\) results in \(f^{-1}(x) = x - 5\).

Step 2: Verifying the Inverse Function: Evaluating \(f(f^{-1}(x))\)

Now that we have our inverse function, we can check the first of two equations, starting with \(f(f^{-1}(x))\). This means we want to substitute \(f^{-1}(x)\) into \(f(x)\), which results in \(f(f^{-1}(x)) = f(x - 5) = (x - 5) + 5 = x\), which verifies the first part.

Step 3: Verifying the Inverse Function: Evaluating \(f^{-1}(f(x))\)

Finally, we want to verify the second equation \(f^{-1}(f(x)) = x\). This means that we want to substitute \(f(x)\) into \(f^{-1}(x)\), resulting in \(f^{-1}(f(x)) = f^{-1}(x + 5) = (x + 5) - 5 = x\). This verifies the second part of the equation.

Key Concepts

One-to-One Functions

One-to-one functions are special types of functions where each input value has a unique output value, and vice versa. This means no two different inputs have the same output. This property is crucial because it allows a function to have an inverse.

Why is one-to-one important?

• An inverse function is only possible if the function is one-to-one. Without this property, the inverse would not be well-defined.
• Graphically, a function is one-to-one if every horizontal line cuts the graph at most once.
• One-to-one functions are also referred to as injective functions.

How do we determine if a function is one-to-one? There are mainly two approaches:

* By the Horizontal Line Test: If no horizontal line intersects the graph more than once, then it is one-to-one.

* Algebraically: If for all pairs of values, whenever the outputs are equal, the inputs must also be equal. So, if \( f(a) = f(b) \), then \( a = b \). This criterion confirms a function is one-to-one.

Function Composition

Function composition is a process of applying one function to the results of another. It's like building a chain where each function feeds its output into the next function in line.

In mathematical terms, if you have two functions, such as \( f(x) \) and \( g(x) \), the composition is represented as \( (f \circ g)(x) = f(g(x)) \).

Why use function composition?

• It helps in creating complex transformations by combining simpler functions.
• Function composition aids in verifying inverse functions.
• It is a foundational concept in calculus and higher mathematics.

When working with inverses, the goal is to show that \( f(f^{-1}(x)) = x \) and \( f^{-1}(f(x)) = x \). This means using function composition to swap a function with its inverse, simplifying down to the identity function.

Verification of Inverses

The verification of inverses is crucial to ensure that the calculated inverse function is correct. It involves checking two main conditions through function composition:

1. \( f(f^{-1}(x)) = x \): Plugging the inverse function into the original function should return the original input \( x \). This checks if the inverse function correctly "undoes" the transformation by \( f(x) \).

2. \( f^{-1}(f(x)) = x \): Substituting \( f(x) \) into the inverse function should also return \( x \). This ensures the inverse function reverses the transformation applied by \( f(x) \).

Verification Techniques:

• Substitute one function into the other systematically.
• Simplify the composition, focusing on both inputs and outputs.
• Look for identity transformations as the final simplified form \( x \).

By satisfying these criteria, we confirm that the inverse function is correct, acting as the "undo" button for the original function.